Deriving the EFT from a Lagrangian with a Right-Handed Neutrino Field

Question

I am working through the MIT EFT course, and I am having trouble with one of the problems. The task is to find the effective field theory (EFT) from the most general Lagrangian with a right-handed neutrino field. The Lagrangian is given by:

$\mathcal{L}_N = i \overline{N} \not\partial N - \frac{M}{2} \overline{N}N^c - \frac{M}{2} \overline{N^c} N + g (\overline{N}H_i \epsilon^{ij} L_j + H_i^*\epsilon^{ij} \overline{L_j} N)$

The solution provided involves rewriting the Lagrangian in terms of the physical field $n=N+N^c$, so that the Lagrangian becomes:

$\mathcal{L}_N = \frac{1}{2}\overline{n} (i \not\partial - M) n + g (\overline{n}H_i \epsilon^{ij} L_j + H_i^*\epsilon^{ij} \overline{L_j} n)$

After some algebraic steps using properties like $\overline{\psi} \xi = \overline{\xi^c} \psi^c$ and $n^c = n$, the Lagrangian is rewritten as:

$\mathcal{L}_N = \frac{1}{2}\overline{n} (i \not\partial - M) n + g \overline{n} (H_i \epsilon^{ij} L_j + H_i^*\epsilon^{ij} L^c_j)$

Then, the equation of motion (EoM) for $\overline{n}$ is computed:

$0 = \frac{\delta \mathcal{L}_N}{\delta \overline{n}} = (i \not\partial - M) n + g (H_i \epsilon^{ij} L_j + H_i^*\epsilon^{ij} L^c_j)$

And thus, $n = -\frac{g}{(i \not\partial - M)}(H_i \epsilon^{ij} L_j + H_i^*\epsilon^{ij} L^c_j)$

And finally substituted into the full Lagrangian to obtain the EFT Lagrangian.

My questions are:

1. When deriving the EoM, shouldn't $n$ and $\overline{n}$ be treated as independent fields? If so, why does the factor $1/2$ disappear when taking the derivative wrt $\overline{n}$? Does the fact that $n = n^c = C\overline{n}^T$ imply that we cannot treat them as independent?

2. The solution substitutes $n$ back into the full Lagrangian to derive the EFT. However, usually when substituting the EoM back into the Lagrangian, it vanishes. This would be the case here if $n$ and $\overline{n}$ were independent fields. In general, do we need to substitute the EoM back to the full Lagrangian? Or only to the interaction terms?

3. I understand that integrating out a heavy field is more rigorously done using path integrals and that using the EoM is typically a tree-level approximation. How would I approach solving this exercise using the path integral formalism?